Draft:Neocategory

Group-like structures

	Total	Associative	Identity	Divisible
Partial magma	Unneeded	Unneeded	Unneeded	Unneeded
Semigroupoid	Unneeded	Required	Unneeded	Unneeded
Small category	Unneeded	Required	Required	Unneeded
Groupoid	Unneeded	Required	Required	Required
Magma	Required	Unneeded	Unneeded	Unneeded
Quasigroup	Required	Unneeded	Unneeded	Required
Unital magma	Required	Unneeded	Required	Unneeded
Loop	Required	Unneeded	Required	Required
Semigroup	Required	Required	Unneeded	Unneeded
Associative quasigroup	Required	Required	Unneeded	Required
Monoid	Required	Required	Required	Unneeded
Group	Required	Required	Required	Required

A neocategory (Ehresmann, who introduced this notion, called it a graphe multiplicatif ) is a generalization of an ordinary category where associative law of composition is weakened to partial magma. Namely, it is the following structure:a one‑to‑one correspondence with the nodes of a directed graph, equipped with partial law of composition that satisfies only left and right identities. As a more general notion, there is the compositional graph, and neocategories can be seen as strongly identitive composition graphs.

Definition

A neocategory $\Sigma$ is couple formed by a set denoted by ${\underline {\Sigma }}$, and a partial law of composition $\kappa$ on ${\underline {\Sigma }}$ satisfying the following axioms:

1. $\kappa$ is a mapping from a subset of ${\underline {\Sigma }}\times {\underline {\Sigma }}$ (denoted by $\Sigma *\Sigma$ and called the set of composable couples) into ${\underline {\Sigma }}$; instead of $\kappa (y,x)$, we write $y\circ x$ and we call $y\circ x$ the composite of $(y,x)$.
2. There exists a graph $({\underline {\Sigma }},\beta ,\alpha )$ (i.e. $\alpha$ and $\beta$ are retractions from ${\underline {\Sigma }}$ onto a subset of ${\underline {\Sigma }}$, denoted by $\Sigma _{0}$), such that:

(existence of units): For each element $x$ of ${\underline {\Sigma }}$, the composites $x\circ \alpha (x)$ and $\beta (x)\circ x$ are defined, and we have

$x\circ \alpha (x)=x=\beta (x)\circ x.$

Here, $\alpha (x)$ is the right identity of $x$ and is called the source of $x$, while $\beta (x)$ is the left identity of $x$ and is called the target of $x$;

(coherence of dom/cod): If the composite $y\circ x$ is defined, then:

$\alpha (y\circ x)=\alpha (x),\ \beta (y\circ x)=\beta (y),\ {\text{and}}\ \alpha (y)=\beta (x).$

From the condition 2, the graph $(\Sigma ,\beta ,\alpha )$ is uniquely defined.

Example

• An ordinary category is a neocategory in which all the couples $(y,x)$ where $\alpha (y)=\beta (x)$ are composable (so that $\Sigma *\Sigma$ is the pullback of $(\alpha ,\beta )$), the law of composition being furthermore associative.

See also

• Sketch (mathematics)
• Partial groupoid
• Non-associative algebra

References

• Bastiani, Andrée; Ehresmann, Charles (1972). "Categories of sketched structures" (PDF). Cahiers de Topologie et Géométrie Différentielle Catégoriques. 13 (2). ISSN 1245-530X.
• Coppey, L. (1980). "Quelques problèmes typiques concernant les graphes multiplicatifs" (PDF). Diagrammes. 3 (2). ISSN 0224-3911.
• Mateus, Paulo; Sernadas, Amílcar; Sernadas, Cristina (1999). "Precategories for Combining Probabilistic Automata". Electronic Notes in Theoretical Computer Science. 29: 169–186. doi:10.1016/S1571-0661(05)80315-9.
• Ehresmann, Charles (1969). "Construction de structures libres". Category Theory, Homology Theory and their Applications II. Lecture Notes in Mathematics. Vol. 92. pp. 74–104. doi:10.1007/BFb0080766. ISBN 978-3-540-04611-0.
• Ehresmann, Charles (1965). Catégories et structures.
• Coppey, L.; Lair, C. (1984). "Leçons de théorie des esquisses" (PDF). Diagrammes. 12 (4). ISSN 0224-3911.

External link

• Tringali, Salvatore (2013). "Plots and Their Applications - Part I: Foundations". arXiv:1311.3524v1 [math.CT].

This page will be placed in the following categories if it is moved to the article namespace.

Categories:

• Algebraic structures
• Category theory
• Non-associative algebra